Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/36

 an arbitrary initial point to the point whose coordinates are $x,$  $y,$  $z,$ is expressed by the integral

$$ \int dw. \sqrt{ \left(\frac{dx}{dw}\right)^{2} + \left(\frac{dy}{dw}\right)^{2} + \left(\frac{dz}{dw}\right)^{2}}$$

If we suppose that the position of the line undergoes an infinitely small variation, so that the coordinates of the different points receive the variations $\delta x,$  $\delta y,$  $\delta z,$ the variation of the whole length becomes

$$\int \frac{dx. d\, \delta x + dy. d\, \delta y + dz. d\, \delta z}{\sqrt{dx^{2} + dy^{2} + dz^{2}}}$$

which expression we can change into the form

$$\begin{array}{c}\displaystyle \frac{dx. \delta x + dy. \delta y + dz. \delta z}    {\sqrt{dx^{2} + dy^{2} + dz^{2}}} \\ \displaystyle -\int \Biggl(   \delta x . d\frac{dx}{\sqrt{dx^{2} + dy^{2} + dz^{2}}}  + \delta y . d\frac{dy}{\sqrt{dx^{2} + dy^{2} + dz^{2}}}  + \delta z . d\frac{dz}{\sqrt{dx^{2} + dy^{2} + dz^{2}}}  \Biggr) \end{array}$$

We know that, in case the line is to be the shortest between its end points, all that stands under the integral sign must vanish. Since the line must lie on the given surface, whose nature is defined by the equation

$$P\, dx + Q\, dy + R\, dz = 0,$$

the variations $\delta x,$  $\delta y,$  $\delta z$ also must satisfy the equation

$$P\, \delta x + Q\, \delta y + R\, \delta z = 0,$$

and from this it follows at once, according to well-known rules, that the differentials

$$d\frac{dx}{\sqrt{dx^{2} + dy^{2} + dz^{2}}},\quad d\frac{dy}{\sqrt{dx^{2} + dy^{2} + dz^{2}}},\quad d\frac{dz}{\sqrt{dx^{2} + dy^{2} + dz^{2}}}$$

must be proportional to the quantities $P,$  $Q,$  $R$ respectively. Let $dr$  be the element of the curved line; $\lambda$  the point on the sphere representing the direction of this element; $L$  the point on the sphere representing the direction of the normal to the curved surface; finally, let $\xi,$  $\eta,$  $\zeta$ be the coordinates of the point $\lambda,$  and $X,$  $Y,$  $Z$  be those of the point $L$  with reference to the centre of the sphere. We shall then have

$$dx = \xi\, dr,\quad dy = \eta\, dr,\quad dz = \zeta\, dr$$

from which we see that the above differentials become $d\xi,$  $d\eta,$  $d\zeta.$ And since the quantities $P,$  $Q,$  $R$  are proportional to $X,$  $Y,$  $Z,$  the character of shortest lines is expressed by the equations

$$\frac{d\xi}{X} = \frac{d\eta}{Y} = \frac{d\zeta}{Z}$$